Pure and Applied Mathematics Journal
Volume 3, Issue 6-2, December 2014, Pages: 1-5
Received: Oct. 8, 2014;
Accepted: Oct. 11, 2014;
Published: Oct. 24, 2014
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Francisco Bulnes, Head of Research Department, Research Department in Mathematics and Engineering, TESCHA, Federal Highway, Mexico-Cuautla Tlapala “La Candelaria”, Chalco, P. C. 56641, Mexico
The integral geometry methods are applied on deformed categories to obtain correspondences in the geometrical Langlands program and construct the due equivalences between geometrical objects of the moduli stacks and algebraic objects of the corresponding categories and their L_(G-opers) characterizing the solution classes to field theory equations in the belonging cohomological context such as H^0 (g[[z] ],V_critical )=C[Op_LG (D^X)] which is natural in the framework of the integral transforms to the generalizing of the Zuckerman functors that will be useful to the obtaining of the different factors of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category H_(G^^ ) ∞ on an adequate moduli stack and the holomorphic L_(G-) bundles category with a special connection (Deligne connection). The corresponding 〖D 〗_-modules may be viewed as sheaves of conformal blocks (or co-invariants) (images under a generalized version of the Penrose transform) naturally arising in the framework of conformal field theory.
Integral Geometry Methods on Deformed Categories in Field Theory II, Pure and Applied Mathematics Journal. Special Issue: Integral Geometry Methods on Derived Categories in the Geometrical Langlands Program.
Vol. 3, No. 6-2,
2014, pp. 1-5.
F. Bulnes, Cohomology of Moduli Spaces in Differential Operators Classification to the Field Theory (II), in: Proceedings of Function Spaces, Differential Operators and Non-linear Analysis., 2011, Tabarz Thur, Germany, Vol. 1 (12) pp001-022.
F. Bulnes, Geometrical Langlands Ramifications and Differential Operators Classification by Coherent D-Modules in Field Theory, Journal of Mathematics and System Sciences, David Publishing, USA Vol. 3, no.10, pp491-507.
F. Bulnes, Penrose Transform on Induced DG/H-Modules and Their Moduli Stacks in the Field Theory, Advances in Pure Mathematics 3 (2) (2013) 246-253. doi: 10.4236/apm.2013.32035.
A. Kapustin, M. Kreuser and K. G. Schlesinger, Homological mirror symmetry: New Developments and Perspectives, Springer. Berlin, Heidelberg, 2009.
F. Bulnes, Integral geometry and complex integral operators cohomology in field theory on space-time, in: Proceedings of 1st International Congress of Applied Mathematics-UPVT (Mexico)., 2009, vol. 1, Government of State of Mexico, pp. 42-51.
F. Bulnes, Penrose Transform on D-Modules, Moduli Spaces and Field Theory, Advances in Pure Mathematics 2 (6) (2012) 379-390. doi: 10.4236/apm.2012.26057.
F. Bulnes, Cohomology of Moduli Spaces on Coherent Sheaves to Conformal Class of the Space-Time, Technical report for XLIII-National Congress of Mathematics of SMM, (RESEARCH) Tuxtla Gutiérrez, Chiapas, Mexico, 2010.
M. Kashiwara and W. Schmid, “Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups, in Lie Theory and Geometry,” Progr. Math. vol. 123, Birkhäuser, Boston, 1994, 457–488.
M. Kashiwara, On the maximally over determined systems of linear differential equation I*. Publ. R.I.M.S. Kyoto Univ. 10 (1975) 563-579.
D. Ben-zvi and D. Nadler, The character theory of complex group, 5 (2011) arXiv:0904.1247v2[math.RT],
D. Ben-Zvi, J. Francis, and D. Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010) 909-966. doi:arXiv:0805.0157.
E. Frenkel, Lectures on the Langlands Program and conformal field theory, Preprint hep-th/0512172.
A. D’Agnolo and P. Schapira. Radon-Penrose transform for D-modules, J. Funct. Anal. 139 (2) (1996) 349–382.
A. Grothendieck, On the De Rham Cohomology of algebraic varieties, Publ. Math.I.H.E.S. 29 (1966) 95-103.
E. Frenkel, Ramifications of the Geometric Langlands Program, CIME Summer School “Representation Theory and Complex Analysis”, Venice, June 2004.
R. Hartshorne, Deformation Theory (Graduate Texts in Mathematics), Springer, USA, 2010.
F. Bulnes (2014) Derived Categories in Langlands Geometrical Ramifications: Approaching by Penrose Transforms. Advances in Pure Mathematics, 4, 253-260. doi: 10.4236/apm.2014.46034.
R. Donagi, T. Pantev, Lectures on the Geometrical Langlands Conjecture and non-Abelian Hodge Theory, 07/01/2008-06/30/2009, , Shing-Tung Yau "Surveys in Differential Geometry 2009", International Press, 2009.
R. Donagi, T. Pantev (2012), Langlands duality for Hitchin systems, Invent.math. arXiv:math/0604617
F. Oort, Yoneda extensions in abelian categories, Mathematische Annalen. 01-1964; 153(3):227-235. DOI: 10.1007/BF01360318
E. Frenkel, C. Teleman, Geometric Langlands correspondence near opers, J. of the Ramanujan Math. Soc. pp. 123–147.
E. Frenkel, C. Teleman, Self extensions of Verma modules and differential forms on opers, Comps. Math. 142 (2006), 477-500.